Prime Numbers

Chapter 8
THE UBIQUITY OF PRIME NUMBERS
It is often remarked that prime numbers finally found a legitimate practical
application in the domain of cryptography. The cryptographic relevance is not
disputed, but there are many other applications of the majestic primes. Some
applications are industrial—such as applications in numerical analysis, applied
mathematics, and other applied sciences—while some are of the “conceptual
feedback” variety, in which primes and their surrounding concepts are used
in theoretical work outside of, say, pure number theory. In this lucrative
research mode, primes are used within algorithms that might appear a priori
independent of primes, and so on. It seems fair to regard the prime number
concept as ubiquitous, since the primes appear in so very many disparate
domains of thought.
8.1 Cryptography
On the face of it, the prime numbers apply to cryptography by virtue of the
extreme difficulty of certain computations. Two such problems are factoring
and the discrete logarithm problem. We shall discuss practical instances of
these problems in the field of cryptography, and also discuss elliptic curve
generalizations.
8.1.1 Diffie–Hellman key exchange
In a monumental paper [Diffie and Hellman 1976], those authors observed
the following “one-way function” behavior of certain group operations. For a
given integer x ≥ 0 and an element g of F ∗ p, the computation of
h = g x
in the field (so, involving continual (mod p) reductions) is generally of
complexity O(ln x) field operations. On the other hand, solving this equation
for x, assuming g, h, p given, is evidently very much harder. As x is an
exponent, and since we are taking something like a logarithm in this
latter problem, the extraction of the unknown x is known as the discrete
logarithm (DL) problem. Though the forward (exponentiation) direction is of
polynomial-time complexity, no general method is known for obtaining the
DL with anything like that efficiency. Some DL algorithms are discussed in
Chapter 5 and in [Schirokauer et al. 1996].